formal series

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11: 3.11 Approximation Techniques

… ►be a formal power series. …

12: 28.31 Equations of Whittaker–Hill and Ince

… ►Formal $2\pi$-periodic solutions can be constructed as Fourier series; compare §28.4: …

13: 2.3 Integrals of a Real Variable

… ►Then the series obtained by substituting (2.3.7) into (2.3.1) and integrating formally term by term yields an asymptotic expansion: …

14: 2.7 Differential Equations

… ►All solutions are analytic at an ordinary point, and their Taylor-series expansions are found by equating coefficients. … ►The radii of convergence of the series (2.7.4), (2.7.6) are not less than the distance of the next nearest singularity of the differential equation from $z_0$. … ►these series converging in an annulus $|z|>a$, with at least one of $f_0$, $g_0$, $g_1$ nonzero. ►Formal solutions are … ►Hence unless the series (2.7.8) terminate (in which case the corresponding ${\mathrm{\Lambda }}_j$ is zero) they diverge. …

15: 27.5 Inversion Formulas

… ►If a Dirichlet series $F(s)$ generates $f(n)$, and $G(s)$ generates $g(n)$, then the product $F(s)G(s)$ generates … ►

27.5.6 $$G(x)=\sum _{n\le x}F\left(\frac{x}{n}\right)⟺F(x)=\sum _{n\le x}\mu \left(n\right)G\left(\frac{x}{n}\right),$$

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Symbols:

$\mu \left(n\right)$: Möbius function, $n$: positive integer, $x$: real number, $F(s)$: Dirichlet series and $G(s)$: Dirichlet series

A&S Ref:

24.3.1 II.C

Permalink:

http://dlmf.nist.gov/27.5.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §27.5 and Ch.27

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27.5.7 $$G(x)=\sum _{m=1}^{\mathrm{\infty }}\frac{F(mx)}{m^s}⟺F(x)=\sum _{m=1}^{\mathrm{\infty }}\mu \left(m\right)\frac{G(mx)}{m^s},$$

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Symbols:

$\mu \left(n\right)$: Möbius function, $m$: positive integer, $x$: real number, $F(s)$: Dirichlet series and $G(s)$: Dirichlet series

Referenced by:

§27.17, §27.5

Permalink:

http://dlmf.nist.gov/27.5.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §27.5 and Ch.27

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16: 16.2 Definition and Analytic Properties

… ►Throughout this chapter it is assumed that none of the bottom parameters $b_1$, $b_2$, $\mathrm{\dots }$, $b_q$ is a nonpositive integer, unless stated otherwise. Then formally … ►When $p\le q$ the series (16.2.1) converges for all finite values of $z$ and defines an entire function. … ►Then the series (16.2.1) terminates and the generalized hypergeometric function is a polynomial in $z$. … ►In general the series (16.2.1) diverges for all nonzero values of $z$. … ►Note also that any partial sum of the generalized hypergeometric series can be represented as a generalized hypergeometric function via …

17: 2.9 Difference Equations

… ►Often $f(n)$ and $g(n)$ can be expanded in series …Formal solutions are … ►For asymptotic expansions in inverse factorial series see Olde Daalhuis (2004a). … ► $c_0=1$, and higher coefficients are determined by formal substitution. … ►The coefficients $b_s$ and constant $c$ are again determined by formal substitution, beginning with $c=1$ when ${\alpha }_2-{\alpha }_1=0$, or with $b_0=1$ when ${\alpha }_2-{\alpha }_1=1,2,3,\mathrm{\dots }$. …

18: 1.12 Continued Fractions

… ►Formally, … ►

Series

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19: Software Index

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	Open Source													With Book						Commercial
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25.21(ix) Dirichlet $L$-series	✓											a	✓									✓
…

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These are collections of software (e.g. libraries) or interactive systems of a somewhat broad scope. Contents may be adapted from research software or may be contributed by project participants who donate their services to the project. The software is made freely available to the public, typically in source code form. While formal support of the collection may not be provided by its developers, within active projects there is often a core group who donate time to consider bug reports and make updates to the collection.

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An increasing number of published books have included digital media containing software described in the book. Often, the collection of software covers a fairly broad area. Such software is typically developed by the book author. While it is not professionally packaged, it often provides a useful tool for readers to experiment with the concepts discussed in the book. The software itself is typically not formally supported by its authors.

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20: 30.4 Functions of the First Kind

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§30.4(iii) Power-Series Expansion

… ►If $f(x)$ is mean-square integrable on $[-1,1]$, then formally …